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  <h1 id="数学-高等数学 第15讲 微分方程" class="content-subhead">数学-高等数学 第15讲 微分方程</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/数学-高等数学 6 第15讲 微分方程.html" class="leancloud_visitors" style="display:none" data-flag-title="数学-高等数学 第15讲 微分方程"></span>
  </p>
  <h2 id="15">第15讲 微分方程</h2>
<blockquote class="content-quote">
<p>微分方程的通解：若微分方程的解中含有任意常数，且任意常数的 <strong>个数</strong> 与微分方程的 <strong>阶数</strong> <strong>相同</strong>，则称之为微分方程的通解。</p>
</blockquote>
<h3 id="1">1. 一阶微分方程的求解</h3>
<h4 id="1_1">（1）换元</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
y'&=f(x)g(y) \\[3ex]
y'&=f(ax+by+c) \ \ \ \ \ \ \ \ \ \ &u=ax+by+c\\[2ex]
y'&=f(\cfrac{y}{x}) &u=\cfrac{y}{x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\[1ex]
\cfrac{1}{y}'&=f(\cfrac{x}{y}) &u=\cfrac{x}{y} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{split}\end{equation}
</script>
</p>
<h4 id="2">（2）齐次线性</h4>
<p>
<script type="math/tex; mode=display">
y'+p(x)y=0 \\[2em]
\underline{方程两边同时乘以：e^{\int{p(x)dx}}} \\[2em]
y = Ce^{-\int{p(x)dx}}
</script>
</p>
<h4 id="3">（3）非齐次线性</h4>
<p>
<script type="math/tex; mode=display">
y'+p(x)y=q(x)
</script>
</p>
<p>
<script type="math/tex; mode=display">
\underline{方程两边同时乘以：e^{\int{p(x)dx}}}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
e^{\int{p(x)dx}}y'+e^{\int{p(x)dx}}p(x)y &= e^{\int{p(x)dx}}q(x) \\[3ex]
                    [e^{\int{p(x)dx}}y]' &= e^{\int{p(x)dx}}q(x) \\[3ex]
                       e^{\int{p(x)dx}}y &= \int{e^{\int{p(x)dx}}q(x)}+C \\[3ex]
                                       y &= e^{-\int{p(x)dx}}[\int{e^{\int{p(x)dx}}q(x)}+C]
\end{split}\end{equation}
</script>
</p>
<h4 id="4">（4）伯努利方程</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
y'+p(x)y &= q(x)y^n \\[2em]
\Rightarrow y^{-n}y'+p(x)y^{1-n} &= q(x) \\[2em]
设\ \ z &= y^{1-n} \\[2ex]
z_x'&= (1-n)y^{-n}y' \\[1ex]
\cfrac{1}{1-n}z_x' &= y^{-n}y' \\[2em]
\Rightarrow \cfrac{1}{1-n}z_x'+p(x)z &= q(x) \\[2em]
\Rightarrow z_x'+(1-n)p(x)z &= (1-n)q(x)
\end{split}\end{equation}
</script>
</p>
<h3 id="2_1">2. 二阶微分方程的求解</h3>
<h4 id="1_2">（1）齐次线性方程的通解</h4>
<p>
<script type="math/tex; mode=display">
y''+py'+qy=0
</script>
</p>
<p>若 <script type="math/tex"> p^2 - 4q \gt 0 </script> ，设 <script type="math/tex"> \lambda_1,\lambda_2 </script> 是特征方程的两个不等实根，即 <script type="math/tex"> \lambda_1\neq\lambda_2 </script> ，可得其通解为<br />
<script type="math/tex; mode=display">
y = C_1e^{\lambda_1x} + C_2e^{\lambda_2x}
</script>
<br />
若 <script type="math/tex"> p^2 - 4q = 0 </script> ，设 <script type="math/tex"> \lambda_1,\lambda_2 </script> 是特征方程的两个相等实根，即二重根，令 <script type="math/tex"> \lambda_1=\lambda_2=\lambda </script> ，可得其通解为<br />
<script type="math/tex; mode=display">
y = (C_1 + C_2x) e^{\lambda x}
</script>
<br />
若 <script type="math/tex"> p^2 - 4q \lt 0 </script> ，设 <script type="math/tex"> \alpha\pm\beta i </script> 是特征方程的一对共轭复根，可得其通解为<br />
<script type="math/tex; mode=display">
y =e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)
</script>
</p>
<h4 id="2-ypyqyfx">（2）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f(x) </script>  的通解</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\big\{y''+py'+qy=f(x)的通解\big\}
= \ \ &\big\{y''+py'+qy=0的通解\big\} \\
+ &\big\{y''+py'+qy=f(x)的特解\big\}
\end{split}\end{equation}
</script>
</p>
<p>求特解的方法：</p>
<p>当 <script type="math/tex">f(x) = P_n(x)</script> 时，设特解为 <script type="math/tex">y^* = Q_n(x)</script>
</p>
<ul>
<li>
<p>比如，<script type="math/tex">P_n(x)=x</script>，则 <script type="math/tex">Q_n(x)=Ax+B</script>
</p>
</li>
<li>
<p>比如，<script type="math/tex">P_n(x)=x^2</script>，则 <script type="math/tex">Q_n(x)=Ax^2+Bx+C</script>
</p>
</li>
</ul>
<p>当 <script type="math/tex"> f(x) = e^{ax}P_n(x) </script> 时，设特解为 <script type="math/tex"> y^{*} = e^{ax}Q_n(x)x^k </script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{cases}
e^{ax}\text{照抄} \\[2ex]
Q_n(x)\text{为x的n次多项式} \\[2ex]
k = 
\begin{cases}
0,\ \alpha\neq\lambda_1,\alpha\ne\lambda_2\\[2ex]
1,\ \alpha=\lambda_1\text{或}\alpha=\lambda_2\\[2ex]
2,\ \alpha=\lambda_1=\lambda_2
\end{cases}
\end{cases}
</script>
</p>
<p>当 <script type="math/tex"> f(x) = e^{ax}[P_m(x)\cos\beta x + P_n(x)\sin\beta x] </script> 时，设特解为 <script type="math/tex"> y^{*} = e^{ax}[Q_l^{(1)}(x)\cos\beta x + Q_l^{(2)}\sin\beta x]x^k </script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{cases}
e^{ax}\text{照抄} \\[2ex]
l=\max\{m,n\} \\[2ex] 
Q_l^{(1)},Q_l^{(2)}\text{为x的两个不同的l次多项式} \\[2ex]
k = 
\begin{cases}
0,\ \alpha\pm\beta \text{i不是特征根} \\[2ex]
1,\ \alpha\pm\beta \text{i是特征根}
\end{cases}
\end{cases}
</script>
</p>
<h4 id="3-ypyqyf_1xf_2x">（3）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f_1(x)+f_2(x)</script>  的通解</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\big\{y''+py'+qy=f_1(x)+f_2(x)的通解\big\}
= \ \ &\big\{y''+py'+qy=0的通解\big\} \\
+ &\big\{y''+py'+qy=f_1(x)的特解\big\} \\
+ &\big\{y''+py'+qy=f_2(x)的特解\big\}
\end{split}\end{equation}
</script>
</p>
<h3 id="3_1">3. 应用</h3>
<h4 id="1_3">（1）曲线切线的斜率</h4>
<p>
<script type="math/tex; mode=display">
f'(x)\bigg|_{x=x_0}=\tan\alpha
</script>
</p>
<h4 id="2_2">（2）面积</h4>
<p>
<script type="math/tex; mode=display">
S = \int_a^b f(x)dx
</script>
</p>
<h4 id="3_2">（3）弧长</h4>
<p>
<script type="math/tex; mode=display">
L = \int_a^b\sqrt{1+(f_x')^2}dx
</script>
</p>
<h4 id="4-x">（4）绕 <script type="math/tex">x</script> 轴旋转体的侧面积</h4>
<p>
<script type="math/tex; mode=display">
S_{绕x轴旋转体侧} = 2\pi r·h = 2\pi\int_a^b\bigg|f(x)\bigg|\ \sqrt{1+(f_x')^2}\ dx
</script>
</p>
<h4 id="5-x">（5）绕 <script type="math/tex">x</script> 轴旋转体的体积</h4>
<p>
<script type="math/tex; mode=display">
V_x = \pi r^2 = \pi\int_a^b f^2(x)dx
</script>
</p>
<h4 id="6-y">（6）绕 <script type="math/tex">y</script> 轴旋转体的体积</h4>
<p>
<script type="math/tex; mode=display">
V_y = 2\pi r·h = 2\pi\int_a^b x\bigg|f(x)\bigg|dx
</script>
</p>
<h4 id="7">（7）平均值</h4>
<p>
<script type="math/tex; mode=display">
\overline f = \cfrac{1}{b-a} \int_a^b f(x)dx = f(\xi)
</script>
</p>
<h4 id="8">（8）曲率</h4>
<p>
<script type="math/tex; mode=display">
k=\cfrac{|f''|}{[1+(f')^2]^{\frac{3}{2}}}
</script>
</p>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 0.75em;" href="#15">第15讲 微分方程</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#1">1. 一阶微分方程的求解</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_1">（1）换元</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2">（2）齐次线性</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3">（3）非齐次线性</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4">（4）伯努利方程</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#2_1">2. 二阶微分方程的求解</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_2">（1）齐次线性方程的通解</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2-ypyqyfx">（2）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f(x) </script>  的通解</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3-ypyqyf_1xf_2x">（3）非齐次线性方程  <script type="math/tex"> y''+py'+qy=f_1(x)+f_2(x)</script>  的通解</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.25em;" href="#3_1">3. 应用</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_3">（1）曲线切线的斜率</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#2_2">（2）面积</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#3_2">（3）弧长</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#4-x">（4）绕 <script type="math/tex">x</script> 轴旋转体的侧面积</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#5-x">（5）绕 <script type="math/tex">x</script> 轴旋转体的体积</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#6-y">（6）绕 <script type="math/tex">y</script> 轴旋转体的体积</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#7">（7）平均值</a>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#8">（8）曲率</a>
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